Directory UMM :Data Elmu:jurnal:S:Socio-Economic Planning Sciences:Vol34.Issue3.2000:
Socio-Economic Planning Sciences 34 (2000) 199±218
www.elsevier.com//locate/dsw
Partitioning input cost eciency into its allocative and
technical components: an empirical DEA application to
hospitals
Jaume Puig-Junoy*
Universitat Pompeu Fabra, Department of Economics and Business, Trias Fargas 25-27, 08005, Barcelona, Spain
Abstract
The study presents an empirical analysis of best practice production and cost frontiers for a sample of
94 acute care hospitals by applying Data Envelopment Analysis (DEA) and a regression model, in a
two-stage approach. This paper contributes to the DEA and eciency measurement literature by adding
results from a homogeneous method of partitioning cost eciency into its allocative (or price) and
technical components, and by decomposing technical eciency into scale, congestion and pure technical
eciency. Allocative eciency is calculated using a DEA assurance approach. It introduces constraints
with lower and upper bounds on the admissible values of weights of the CCD DEA model that
computes technical eciency. We thus obtain scores unbiased by the lack of precise information on
input prices. In the second stage, a log-regression model is employed to test a number of hypotheses
involving the role of ownership, market structure, and regulation in terms of dierences amongst the
various eciency concepts measured. Results highlight the relevance of market concentration and public
®nance in explaining these dierences. 7 2000 Elsevier Science Ltd. All rights reserved.
Keywords: Hospital performance; Cost eciency; Allocative eciency; Technical eciency; Data envelopment
analysis; Assurance region
1. Introduction
The purpose of this paper is to obtain empirical and complementary measures of hospital
* Corresponding author. Fax: +34-3-542-17-46.
E-mail address: [email protected] (J. Puig-Junoy).
0038-0121/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 3 8 - 0 1 2 1 ( 9 9 ) 0 0 0 2 4 - 5
200
J. Puig-Junoy / Socio-Economic Planning Sciences 34 (2000) 199±218
performance rooted in the principles of production economics, and to evaluate the factors that
contribute to performance. The method is applied to 94 acute care hospitals operating in the
context of a National Health Service in Catalonia (Spain)1. Assessing performance is a
necessary step in the design and implementation of privatization of ownership and
management policies, and in fostering competition and other deregulating measures in health
and hospital services. In this regard, health care purchasers in all systems are now seeking ways
to improve hospital eciency.
Hospital performance is proxied in this paper using measures of Farrell's [1] de®nition of
technical and allocative eciency. These are partial, but theoretically rooted, indicators of
hospital performance. A hospital is said to be technically ecient if a reduction in any input
requires an increase in at least one other input or a decrease in at least one output. A hospital
is allocatively inecient if it does not select the optimal mix of inputs given the available
technology and the input prices it faces. Technical eciency has been advocated as a measure
to compare performance of ®rms having dierent ownership regimes or legal statuses. It is
particularly useful in evaluating the performance of public sector and nonpro®t activities,
which are predominant in the hospital sector. Technical eciency may be achieved
independently of allocative eciency.
1.1. Measuring hospital eciency
Empirical measurement of ineciency has been accomplished using two classes of
methodologies: stochastic parametric regression-based methods and nonstochastic
nonparametric mathematical programming methods. Data envelopment analysis (DEA) is the
most used family of linear programming models.
A number of papers have measured hospital eciency on the basis of the best-practice
frontier by using both methodologies. Ineciency provided by hospital cost frontiers is the
result of technical and allocative ineciency combinations in unknown proportions [2]. Eakin
[3] is an exception, computing allocative eciency scores. Some advances in frontier regression
analysis allow one to obtain dierentiated measures of technical and allocative ineciencies by
introducing restrictions equalizing marginal productivity ratios and price ratios in the cost
function. Nevertheless, computational diculties in panel data aside, some problems exist in
ruling out X-ineciency when separating both types of ineciency in cost frontier regression
analysis. These are due to the assumption that maximizing behaviour is present [4] since it uses
the so-called Shepard cost share equations to estimate model parameters. In response to this
situation, several DEA models are proposed here to partition cost eciency into its allocative
and technical components within a multiple input multiple output production process.
An increasing number of researchers have applied DEA to hospital eciency analysis. Some
recent examples include: Burgess and Wilson [5], Valdmanis [6], Ozcan and Luke [7],
Magnussen [8], and Dalmau and Puig-Junoy [9]. The hospital DEA literature has restricted its
1
Catalonia is a region with six million inhabitants. The hospital system in Catalonia may be summarized as acting
in a National Health Service with the following measures: 99% of population with public insurance, 73% public
®nancing and 39% public production
J. Puig-Junoy / Socio-Economic Planning Sciences 34 (2000) 199±218
201
attention to technical eciency, although cost-minimizing eciency includes both technical and
allocative eciency. To our knowledge, only two papers calculate hospital allocative eciency
[10,11] using nonparemetric models. Calculation of allocative eciency requires accurate
information on prices of inputs. Morey et al. [10] and Byrnes and Valdmanis [11] use average
prices to calculate allocative eciency for public and nonpro®t hospitals in California in the
period 1982±83. However, in both cases, average input prices involve an unreasonably wide
range of variation between hospitals, which is not justi®ed by the authors. In this regard, we
suggest that less quality of cost data are available than physical input data in self-reported
sources of information. As might be expected, a major diculty is encountered in securing the
price information needed to implement the concept of allocative eciency.
This paper's contribution to the DEA applications literature involves the use of this method
to derive both allocative and technical eciency scores for hospitals, thus overcoming the
traditional con®nement to technical eciency in earlier eorts. The use of DEA provides the
opportunity to partition cost eciency into its allocative and technical eciency (and the latter
into pure technical, congestion and scale eciency), and to subsequently obtain comparable
measures of the dierent theoretical eciency concepts. Results should cast light on the relative
importance of the dierent types of ineciency for hospitals under analysis. Additionally, a
DEA assurance approach is applied to the calculation of allocative eciency in order to obtain
scores unbiased by the lack of precise information on input prices.
1.2. Explaining variations in hospital measured eciency
According to Pestieau and Tulkens' [12] theoretical and empirical revision, three categories
of factors might be distinguished in assessing and explaining the performance of public and
nonpro®t enterprises: ownership (and ®rm objectives), competition, and regulation. In order to
assess the expected eects of projected and in-course hospital policies, it is thus of crucial
importance to ascertain the potential impact of ownership, market structure and regulation on
the explanation of dierences in eciency scores.
Evidence from empirical analyses of hospital ineciency using DEA several times on the
same set of data, Grosskopf and Valdmanis [13] and Valdmanis [6] suggest that public
hospitals are more technically ecient than are nonpro®t and private ones. Register and
Brunning [14], also using DEA, found no dierences between nonpro®t and public hospitals
when comparing technical eciency. Ozcan et al. [15] and Ozcan and Luke [7] observed that
US government hospitals tend to be more ecient, and for-pro®t hospitals less ecient, than
other hospitals. Chirikos and Sear [16] conclude that for-pro®t hospitals are technically less
ecient when they perform in less competitive markets.
A number of earlier papers have documented a positive relation between costs per admission
or per patient-day and more competitive markets [e.g. 17,18], usually attributed to the eects
of nonprice competition. Nevertheless, only a few of these papers have addressed the relation
between eciency and competition. Recent empirical research estimating a frontier cost
function found weak evidence to sustain the notion that competition from other hospitals is
related to ineciency [2, 3]. A positive relation between competition and higher average cost or
cost ineciency does not necessarily imply technical ineciency. It might, for example, be a
case of exclusively allocative ineciency, or both technical and allocative ineciency in
202
J. Puig-Junoy / Socio-Economic Planning Sciences 34 (2000) 199±218
dierent, unknown proportions. Two studies explicitly address the eect of competition on
technical ineciency by explaining dierences in DEA scores. Register and Brunning [14] did
not ®nd any relation between DEA scores and market concentration. Chirikos and Sear's [16]
results showed that ineciency scores are higher in markets with more vigorous inter-hospital
competition, the relation being more intense in highly competitive markets.
Analysis of the relation between regulation and hospital eciency has focused nearly
exclusively on Medicare Prospective Payment System evidence (PPS). Zuckerman et al. [2] thus
found that pro®t rates are signi®cantly higher among relatively less cost inecient hospitals
subject to PPS. Chirikos and Sear [16] found no signi®cant relation between technical eciency
and an index of early PPS pressures.
In this paper, an evaluation of the eects of observed present market structure, ownership
and regulation on hospital allocative and pure technical eciency for 94 Catalan acute care
hospitals is developed. In reference to the relation between hospital performance and factors
explaining performance, this paper adds to the preceding literature in three aspects. Firstly, it
does not restrict attention to larger or urban hospitals since all acute care hospitals are
considered. It uses a Her®ndahl±Hirschman index [19] of concentration calculated for every
hospital using patient origin data, and it expands evidence to hospitals in a European National
Health Service context. Secondly, it encompasses the analysis of a wider range of
environmental variables considered simultaneously as factors explaining eciency, and it also
considers some control variables for eciency scores. Ratios partially measuring ineciency
are ruled out as factors explaining eciency (i.e. occupancy rate, length of stay, etc.). And,
thirdly, it sheds separate light on the eects of environmental variables on allocative and
technical eciency (rather than on average production/cost functions).
The paper is organized as follows. Section 2 lays out the general framework for the
application of Data Envelopment Analysis to the measurement of cost and technical eciency.
Variable de®nitions and descriptions are presented in Section 3. Section 4 presents DEA
allocative and technical ineciency results. A regression analysis of the DEA eciency scores
is presented in Section 5, while section 6 concludes.
2. The performance evaluation methodology
As noted previously, hospital performance is proxied in this paper by allocative and
technical eciency. In this section, we provide de®nitions of eciency used and their methods
of measurement.
2.1. Eciency de®nition
To characterize production technology relative to which eciency is measured, each hospital
uses variable inputs x=(x1,. . . ,xN ) $ RN+ to produce variable outputs y=( y1,. . . ,yM ) $ RM+.
Inputs are transformed into outputs using a technology that can be described by the graph
GR={(x,y ): x can produce y }. Corresponding to the graph, there is a family of input sets
L( y )={x:(x,y ) $ GR}, y $ RM+. Input sets are assumed to be closed and bounded above, and
to satisfy strong disposability of inputs. The input sets contain isoquants Isoq L( y )={x:x $
J. Puig-Junoy / Socio-Economic Planning Sciences 34 (2000) 199±218
203
L( y ),yx $ L( y ),y ( (0,1)}, y $ RM+. Also corresponding to the graph of the technology is a
family of output sets P(x )={ y:( y,x ) $ GR},x $ RN+. Output sets are assumed to be closed and
bounded above, and to satisfy the properties of convexity and strong disposability of outputs.
A Farrell±Debreu radial measure of the technical eciency of input vector x in the
production of output vector y is given by: TE(x,y )=min {y:yx $ L( y )}, where y=1 indicates
radial technical eciency and y < 1 shows the degree of radial technical ineciency. According
to Farrell's concept, the cost eciency of a hospital using input vector x to produce output
vector y when input prices are w is measured by the ratio of minimum cost to actual cost: CE
(x,y,w )=c( y,w )/w Tx, where c( y,w ) is the cost function (the minimum expenditure required to
produce y when input prices are w ), where CE (x,y,w )=1 indicates cost eciency and CE
(x,y,w ) < 1 shows the degree of cost ineciency.
2.2. Eciency measurement
Assuming strong input and output disposability, the input cost eciency measure (CE)
may be decomposed into its input allocative eciency (AE), scale eciency (SE), input
congestion (C), and pure technical eciency (PTE) components [20, p. 80]: CE
(x,y,w )=AE(x,y,w )SE(x,y )C(x,y )PTE(x,y ). As FaÈre et al. [20] state ``the input cost
ineciency must be due to selection of the wrong input mix, to the adoption of an ineciently
small or large scale, to input congestion, or to purely technical ineciency''. Scale ineciency
thus occurs because the hospital is not operating at the scale of operation consistent with longrun competitive equilibrium. Also, technical eciency (TE) is de®ned as the product of the
scale eciency, input congestion, and pure technical eciency components:
TE(x,y )=SE(x,y )C(x,y )PTE(x,y ). The Farrell input allocative eciency of a hospital is
measured as the ratio of cost eciency to overall technical eciency: AE(x,y,w )=CE(x,y,w )/
TE(x,y ), where AE (x,y,w )=1 indicates input allocative eciency and AE (x,y,w ) < 1 shows
the degree of input allocative ineciency.
Assume the hospital under evaluation as having data (x 0,y 0,w 0), and consider the inputoriented CCR DEA model [21] in the primal (envelopment) formulation, where xi $ RN+ and yi
$ RM+, and i = 1. . . I, where I indicates the number of hospitals in the sample:
TE x 0 , y0 min y
y, l
subject to
yx 0 ÿ Xle0
ÿy0 Yle0
le0
where X is an N I input matrix with columns Xi, Y is an M I output matrix with columns
yi, and l is an i 1 intensity vector. The optimal value of y provides a technical eciency
measure of the hospital under evaluation. Input-oriented radial eciency requires u Ty8=y=1.
204
J. Puig-Junoy / Socio-Economic Planning Sciences 34 (2000) 199±218
A hospital is judged to be technically inecient if, at optimum, y < 1, and technically ecient
if, at optimum, y=1. The input-oriented CCR DEA model incorporates the assumption of
constant returns to scale in production.
Banker et al. [22] (BCC) generalized the CCR formulation to allow variable returns to scale.
The input-oriented BCC DEA model computes, exclusively, a pure technical eciency measure
(W ) by introducing an additional restriction to the input-oriented CCR DEA model: e Tl=1,
where e T is an I 1 row vector of ones. This pure technical eciency measure is obtained
under the restriction of weak input disposability but allows for variable returns to scale. The
above decomposition of input cost eciency requires PTE to be computed by relaxing the
strong input disposability restriction, to allow for an input congestion component. The
congestion component is due to production on a backward-bending segment of the isoquant
that is in the region where marginal product is negative. Pure technical eciency with weak
disposability of inputs may be computed from the following problem in the primal
(envelopment) formulation:
PTE x 0 , y0 min y
y, l, s
subject to
ysx 0 ÿ Xl 0
ÿy0 Yle0
eT l 1
0
www.elsevier.com//locate/dsw
Partitioning input cost eciency into its allocative and
technical components: an empirical DEA application to
hospitals
Jaume Puig-Junoy*
Universitat Pompeu Fabra, Department of Economics and Business, Trias Fargas 25-27, 08005, Barcelona, Spain
Abstract
The study presents an empirical analysis of best practice production and cost frontiers for a sample of
94 acute care hospitals by applying Data Envelopment Analysis (DEA) and a regression model, in a
two-stage approach. This paper contributes to the DEA and eciency measurement literature by adding
results from a homogeneous method of partitioning cost eciency into its allocative (or price) and
technical components, and by decomposing technical eciency into scale, congestion and pure technical
eciency. Allocative eciency is calculated using a DEA assurance approach. It introduces constraints
with lower and upper bounds on the admissible values of weights of the CCD DEA model that
computes technical eciency. We thus obtain scores unbiased by the lack of precise information on
input prices. In the second stage, a log-regression model is employed to test a number of hypotheses
involving the role of ownership, market structure, and regulation in terms of dierences amongst the
various eciency concepts measured. Results highlight the relevance of market concentration and public
®nance in explaining these dierences. 7 2000 Elsevier Science Ltd. All rights reserved.
Keywords: Hospital performance; Cost eciency; Allocative eciency; Technical eciency; Data envelopment
analysis; Assurance region
1. Introduction
The purpose of this paper is to obtain empirical and complementary measures of hospital
* Corresponding author. Fax: +34-3-542-17-46.
E-mail address: [email protected] (J. Puig-Junoy).
0038-0121/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 3 8 - 0 1 2 1 ( 9 9 ) 0 0 0 2 4 - 5
200
J. Puig-Junoy / Socio-Economic Planning Sciences 34 (2000) 199±218
performance rooted in the principles of production economics, and to evaluate the factors that
contribute to performance. The method is applied to 94 acute care hospitals operating in the
context of a National Health Service in Catalonia (Spain)1. Assessing performance is a
necessary step in the design and implementation of privatization of ownership and
management policies, and in fostering competition and other deregulating measures in health
and hospital services. In this regard, health care purchasers in all systems are now seeking ways
to improve hospital eciency.
Hospital performance is proxied in this paper using measures of Farrell's [1] de®nition of
technical and allocative eciency. These are partial, but theoretically rooted, indicators of
hospital performance. A hospital is said to be technically ecient if a reduction in any input
requires an increase in at least one other input or a decrease in at least one output. A hospital
is allocatively inecient if it does not select the optimal mix of inputs given the available
technology and the input prices it faces. Technical eciency has been advocated as a measure
to compare performance of ®rms having dierent ownership regimes or legal statuses. It is
particularly useful in evaluating the performance of public sector and nonpro®t activities,
which are predominant in the hospital sector. Technical eciency may be achieved
independently of allocative eciency.
1.1. Measuring hospital eciency
Empirical measurement of ineciency has been accomplished using two classes of
methodologies: stochastic parametric regression-based methods and nonstochastic
nonparametric mathematical programming methods. Data envelopment analysis (DEA) is the
most used family of linear programming models.
A number of papers have measured hospital eciency on the basis of the best-practice
frontier by using both methodologies. Ineciency provided by hospital cost frontiers is the
result of technical and allocative ineciency combinations in unknown proportions [2]. Eakin
[3] is an exception, computing allocative eciency scores. Some advances in frontier regression
analysis allow one to obtain dierentiated measures of technical and allocative ineciencies by
introducing restrictions equalizing marginal productivity ratios and price ratios in the cost
function. Nevertheless, computational diculties in panel data aside, some problems exist in
ruling out X-ineciency when separating both types of ineciency in cost frontier regression
analysis. These are due to the assumption that maximizing behaviour is present [4] since it uses
the so-called Shepard cost share equations to estimate model parameters. In response to this
situation, several DEA models are proposed here to partition cost eciency into its allocative
and technical components within a multiple input multiple output production process.
An increasing number of researchers have applied DEA to hospital eciency analysis. Some
recent examples include: Burgess and Wilson [5], Valdmanis [6], Ozcan and Luke [7],
Magnussen [8], and Dalmau and Puig-Junoy [9]. The hospital DEA literature has restricted its
1
Catalonia is a region with six million inhabitants. The hospital system in Catalonia may be summarized as acting
in a National Health Service with the following measures: 99% of population with public insurance, 73% public
®nancing and 39% public production
J. Puig-Junoy / Socio-Economic Planning Sciences 34 (2000) 199±218
201
attention to technical eciency, although cost-minimizing eciency includes both technical and
allocative eciency. To our knowledge, only two papers calculate hospital allocative eciency
[10,11] using nonparemetric models. Calculation of allocative eciency requires accurate
information on prices of inputs. Morey et al. [10] and Byrnes and Valdmanis [11] use average
prices to calculate allocative eciency for public and nonpro®t hospitals in California in the
period 1982±83. However, in both cases, average input prices involve an unreasonably wide
range of variation between hospitals, which is not justi®ed by the authors. In this regard, we
suggest that less quality of cost data are available than physical input data in self-reported
sources of information. As might be expected, a major diculty is encountered in securing the
price information needed to implement the concept of allocative eciency.
This paper's contribution to the DEA applications literature involves the use of this method
to derive both allocative and technical eciency scores for hospitals, thus overcoming the
traditional con®nement to technical eciency in earlier eorts. The use of DEA provides the
opportunity to partition cost eciency into its allocative and technical eciency (and the latter
into pure technical, congestion and scale eciency), and to subsequently obtain comparable
measures of the dierent theoretical eciency concepts. Results should cast light on the relative
importance of the dierent types of ineciency for hospitals under analysis. Additionally, a
DEA assurance approach is applied to the calculation of allocative eciency in order to obtain
scores unbiased by the lack of precise information on input prices.
1.2. Explaining variations in hospital measured eciency
According to Pestieau and Tulkens' [12] theoretical and empirical revision, three categories
of factors might be distinguished in assessing and explaining the performance of public and
nonpro®t enterprises: ownership (and ®rm objectives), competition, and regulation. In order to
assess the expected eects of projected and in-course hospital policies, it is thus of crucial
importance to ascertain the potential impact of ownership, market structure and regulation on
the explanation of dierences in eciency scores.
Evidence from empirical analyses of hospital ineciency using DEA several times on the
same set of data, Grosskopf and Valdmanis [13] and Valdmanis [6] suggest that public
hospitals are more technically ecient than are nonpro®t and private ones. Register and
Brunning [14], also using DEA, found no dierences between nonpro®t and public hospitals
when comparing technical eciency. Ozcan et al. [15] and Ozcan and Luke [7] observed that
US government hospitals tend to be more ecient, and for-pro®t hospitals less ecient, than
other hospitals. Chirikos and Sear [16] conclude that for-pro®t hospitals are technically less
ecient when they perform in less competitive markets.
A number of earlier papers have documented a positive relation between costs per admission
or per patient-day and more competitive markets [e.g. 17,18], usually attributed to the eects
of nonprice competition. Nevertheless, only a few of these papers have addressed the relation
between eciency and competition. Recent empirical research estimating a frontier cost
function found weak evidence to sustain the notion that competition from other hospitals is
related to ineciency [2, 3]. A positive relation between competition and higher average cost or
cost ineciency does not necessarily imply technical ineciency. It might, for example, be a
case of exclusively allocative ineciency, or both technical and allocative ineciency in
202
J. Puig-Junoy / Socio-Economic Planning Sciences 34 (2000) 199±218
dierent, unknown proportions. Two studies explicitly address the eect of competition on
technical ineciency by explaining dierences in DEA scores. Register and Brunning [14] did
not ®nd any relation between DEA scores and market concentration. Chirikos and Sear's [16]
results showed that ineciency scores are higher in markets with more vigorous inter-hospital
competition, the relation being more intense in highly competitive markets.
Analysis of the relation between regulation and hospital eciency has focused nearly
exclusively on Medicare Prospective Payment System evidence (PPS). Zuckerman et al. [2] thus
found that pro®t rates are signi®cantly higher among relatively less cost inecient hospitals
subject to PPS. Chirikos and Sear [16] found no signi®cant relation between technical eciency
and an index of early PPS pressures.
In this paper, an evaluation of the eects of observed present market structure, ownership
and regulation on hospital allocative and pure technical eciency for 94 Catalan acute care
hospitals is developed. In reference to the relation between hospital performance and factors
explaining performance, this paper adds to the preceding literature in three aspects. Firstly, it
does not restrict attention to larger or urban hospitals since all acute care hospitals are
considered. It uses a Her®ndahl±Hirschman index [19] of concentration calculated for every
hospital using patient origin data, and it expands evidence to hospitals in a European National
Health Service context. Secondly, it encompasses the analysis of a wider range of
environmental variables considered simultaneously as factors explaining eciency, and it also
considers some control variables for eciency scores. Ratios partially measuring ineciency
are ruled out as factors explaining eciency (i.e. occupancy rate, length of stay, etc.). And,
thirdly, it sheds separate light on the eects of environmental variables on allocative and
technical eciency (rather than on average production/cost functions).
The paper is organized as follows. Section 2 lays out the general framework for the
application of Data Envelopment Analysis to the measurement of cost and technical eciency.
Variable de®nitions and descriptions are presented in Section 3. Section 4 presents DEA
allocative and technical ineciency results. A regression analysis of the DEA eciency scores
is presented in Section 5, while section 6 concludes.
2. The performance evaluation methodology
As noted previously, hospital performance is proxied in this paper by allocative and
technical eciency. In this section, we provide de®nitions of eciency used and their methods
of measurement.
2.1. Eciency de®nition
To characterize production technology relative to which eciency is measured, each hospital
uses variable inputs x=(x1,. . . ,xN ) $ RN+ to produce variable outputs y=( y1,. . . ,yM ) $ RM+.
Inputs are transformed into outputs using a technology that can be described by the graph
GR={(x,y ): x can produce y }. Corresponding to the graph, there is a family of input sets
L( y )={x:(x,y ) $ GR}, y $ RM+. Input sets are assumed to be closed and bounded above, and
to satisfy strong disposability of inputs. The input sets contain isoquants Isoq L( y )={x:x $
J. Puig-Junoy / Socio-Economic Planning Sciences 34 (2000) 199±218
203
L( y ),yx $ L( y ),y ( (0,1)}, y $ RM+. Also corresponding to the graph of the technology is a
family of output sets P(x )={ y:( y,x ) $ GR},x $ RN+. Output sets are assumed to be closed and
bounded above, and to satisfy the properties of convexity and strong disposability of outputs.
A Farrell±Debreu radial measure of the technical eciency of input vector x in the
production of output vector y is given by: TE(x,y )=min {y:yx $ L( y )}, where y=1 indicates
radial technical eciency and y < 1 shows the degree of radial technical ineciency. According
to Farrell's concept, the cost eciency of a hospital using input vector x to produce output
vector y when input prices are w is measured by the ratio of minimum cost to actual cost: CE
(x,y,w )=c( y,w )/w Tx, where c( y,w ) is the cost function (the minimum expenditure required to
produce y when input prices are w ), where CE (x,y,w )=1 indicates cost eciency and CE
(x,y,w ) < 1 shows the degree of cost ineciency.
2.2. Eciency measurement
Assuming strong input and output disposability, the input cost eciency measure (CE)
may be decomposed into its input allocative eciency (AE), scale eciency (SE), input
congestion (C), and pure technical eciency (PTE) components [20, p. 80]: CE
(x,y,w )=AE(x,y,w )SE(x,y )C(x,y )PTE(x,y ). As FaÈre et al. [20] state ``the input cost
ineciency must be due to selection of the wrong input mix, to the adoption of an ineciently
small or large scale, to input congestion, or to purely technical ineciency''. Scale ineciency
thus occurs because the hospital is not operating at the scale of operation consistent with longrun competitive equilibrium. Also, technical eciency (TE) is de®ned as the product of the
scale eciency, input congestion, and pure technical eciency components:
TE(x,y )=SE(x,y )C(x,y )PTE(x,y ). The Farrell input allocative eciency of a hospital is
measured as the ratio of cost eciency to overall technical eciency: AE(x,y,w )=CE(x,y,w )/
TE(x,y ), where AE (x,y,w )=1 indicates input allocative eciency and AE (x,y,w ) < 1 shows
the degree of input allocative ineciency.
Assume the hospital under evaluation as having data (x 0,y 0,w 0), and consider the inputoriented CCR DEA model [21] in the primal (envelopment) formulation, where xi $ RN+ and yi
$ RM+, and i = 1. . . I, where I indicates the number of hospitals in the sample:
TE x 0 , y0 min y
y, l
subject to
yx 0 ÿ Xle0
ÿy0 Yle0
le0
where X is an N I input matrix with columns Xi, Y is an M I output matrix with columns
yi, and l is an i 1 intensity vector. The optimal value of y provides a technical eciency
measure of the hospital under evaluation. Input-oriented radial eciency requires u Ty8=y=1.
204
J. Puig-Junoy / Socio-Economic Planning Sciences 34 (2000) 199±218
A hospital is judged to be technically inecient if, at optimum, y < 1, and technically ecient
if, at optimum, y=1. The input-oriented CCR DEA model incorporates the assumption of
constant returns to scale in production.
Banker et al. [22] (BCC) generalized the CCR formulation to allow variable returns to scale.
The input-oriented BCC DEA model computes, exclusively, a pure technical eciency measure
(W ) by introducing an additional restriction to the input-oriented CCR DEA model: e Tl=1,
where e T is an I 1 row vector of ones. This pure technical eciency measure is obtained
under the restriction of weak input disposability but allows for variable returns to scale. The
above decomposition of input cost eciency requires PTE to be computed by relaxing the
strong input disposability restriction, to allow for an input congestion component. The
congestion component is due to production on a backward-bending segment of the isoquant
that is in the region where marginal product is negative. Pure technical eciency with weak
disposability of inputs may be computed from the following problem in the primal
(envelopment) formulation:
PTE x 0 , y0 min y
y, l, s
subject to
ysx 0 ÿ Xl 0
ÿy0 Yle0
eT l 1
0